# Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}film [27] were studied. For the chiral magnet Cu

_{2}OSeO

_{3}, the metastable skyrmion lattice emerged under electric fields [28]. As for the theoretical investigations conducted using ab initio density functional theory, the effects of electric fields on the phase boundaries, crystal growth rates, nucleation rates, and interfacial free energies have been studied [29,30]. The electric-field-tuned topological phase transition was investigated by first-principles calculations [31,32,33], with a change in the dipole moment [34]. Moreover, resistive switching in phase change materials [35] has recently attracted considerable attention for its application in non-volatile memory devices, and the magnetic properties have been studied [36]. Despite the extensive investigations being carried out, it remains poorly understood how the molecules aggregate and organize into a new form under external electric fields [1,3,4,5,6]. The main reason is that it exceeds the range of instrument testing for small molecules in experiments, and in theoretical simulations, the time scale is many orders of magnitude lower than that of real occurrence [37,38]. In fact, the nucleation itself is a rare event [39,40], and it is very difficult to find a set of suitable collective variables that can describe the reaction coordinate [39], which is the challenge of nucleation simulation and understanding nucleation mechanisms at the molecular level [41].

## 2. Theory

#### 2.1. Order Parameters

_{ij}is the distance between the centers-of-mass of two molecules; ${\varphi}_{\widehat{r}}$ is defined as the bond orientation from the projection of ${\widehat{r}}_{ij}$ onto the absolute orientation of the ith molecule; ${\varphi}_{q}$ is the relative orientation resulting from the rotation of the absolute orientation of the ith molecule onto jth molecule; and r

_{α}, ${\varphi}_{\widehat{r}}^{\alpha}$, and ${\varphi}_{q}^{\alpha}$ are the mean center-of-mass distance, mean bond orientation, and mean relative orientation corresponding to the α-peak, respectively. ${\sigma}_{\alpha}$, ${\eta}_{\widehat{r}}^{\alpha}$, and ${\eta}_{q}^{\alpha}$ are the standard deviation and concentration parameters, respectively, and I

_{0}is the modified Bessel function of the second kind and order 0. Thus, the OPs of the bond distance combined with the bond orientation or relative orientation could be given by

#### 2.2. Finite-Temperature String

_{1}(x), θ

_{2}(x), …, θ

_{N}(x)), where x ∈ R

^{n}are the Cartesian coordinates, the function of the free energy depending on z = (z

_{1}, z

_{2}, …, z

_{N}) is defined as

_{B}T, where k

_{B}and T are the Boltzmann constant and the temperature, respectively. Note that N is less than the dimensionality of the full system.

_{ij}(α, t) is the projector on the plane perpendicular to the path at z(α, t), M

_{jk}(z(α, t)) is the tensor, and ${\scriptscriptstyle \frac{\partial F\left(z\left(\alpha ,t\right)\right)}{\partial {z}_{k}}}$ (i.e., $\nabla F(z)$) is the free-energy gradient (i.e., mean force). Given two minima of the free energy located at z

_{a}and z

_{b}, Equation (9) can be solved according to the boundary conditions z(0, t) = z

_{a}and z(1, t) = z

_{b}. As t→∞ the solution of (9) converges to an MFEP connecting z

_{a}and z

_{b}, and simultaneously, the tangent vector is parallel to $M(z)\nabla F(z)$ at every point z, i.e.,

_{jk}(z) is defined as

#### 2.3. Minimum-Free-Energy Path from Finite-Temperature String

_{i}is given by

_{i}, s

_{i}, and a

_{i}represent the density of sample point i (i.e., the number of points i within a certain range), the distance between different clusters or datasets, and the distance between different sample points in the same cluster, respectively. They can be calculated by Equations (17) and (18) as follows:

_{,}$averge\left(m-1,m+1\right)=\left({\theta}_{C,m-1}+{\theta}_{C,m+1}\right)/2$. The smoothness of the temporary string was calculated by the Monte Carlo method. Firstly, the smoothing score of the current temporary string was calculated. Then, in the same space, another clustering center, where the weight difference from the first clustering center is small, was selected, and the smoothing score was recalculated. According to the new smoothing score, whether to accept the new change was determined according to the metropolis criterion. Finally, a string with the lowest smoothing score was obtained.

## 3. Results and Discussion

#### 3.1. Crystallization of TKX-50 by DC External Electric Field

_{1}/c, the crystal data and structural parameters are changed, obviously. In particular, the volume decreased, and the density increased by 0.1 g/cm

^{3}. For the explosive, the higher the density, the higher the detonation velocity and pressure become. Therefore, the introduction of the external electric field into the explosive system is beneficial for improving the performance of explosives. Moreover, from Tables S3 and S8, it can be seen that the bond length of C–N on the TKX-50 tetrazole ring is shortened by an average of 0.04 Å under the external electric field, suggesting that the external electric field enhances the stability of TKX-50.

#### 3.2. MD Simulation on Crystallization of TKX-50 without External Electric Field

#### 3.2.1. Peaks in Pair Distribution Function

^{+}][Cl

^{−}] from its supercooled liquid phase, two [dmim

^{+}][Cl

^{−}] ion pairs show the variables relevant for the construction of OPs, which are based on the [dmim

^{+}] cations. The vector r joins the center of mass of the two cations. The vector normal to the plane of the imidazolium ring gives the absolute orientation q of each of the cations [71]. Similarly, according to the construction method of OPs by Santiso et al. [39], the vector r joins the center of mass of two tetrazolium anion rings, and the absolute orientation q was given by two anions. The bond orientation ${\varphi}_{\widehat{r}}$ and the relative orientation ${\varphi}_{q}$ were also given by two anions; see “Section 4.2”. Thus, the local OPs of the TKX-50 crystal and the corresponding forms in the formic acid solution were constructed by the structural variables involving the bond distances, bond orientations, and relative orientations (see Figure 3). After obtaining the maximum-likelihood estimators, they were calculated (see Figure 4). Although three OPs of the TKX-50 crystal phase and the corresponding forms in the formic acid solution overlap slightly, the peaks exhibit significantly different values. Therefore, any of the OPs can serve as a good metric to detect the orders of the crystalline and the corresponding forms in the formic acid solution. In this work, we chose the bond orientation OPs and relative orientation OPs during all our simulations.

#### 3.2.2. Convergence of FTS and K-Means Clustering

#### 3.2.3. Minimum-Free-Energy Path

^{−1}in all the cases, while the difference in PMF between the TKX-50 crystal and the transition state is large, close to 50.0 kJ∙mol

^{−1}from ${\varphi}_{C}^{db}$ in two cases and ${\varphi}_{C}^{dr}$ as the collective variables by K-means clustering, and about 40.0 kJ∙mol

^{−1}from ${\varphi}_{C}^{dr}$ without K-means clustering. These results show that, at 300 K, the solubility of TKX-50 in formic acid is not high, while its saturated solution is prone to crystallization. This is in agreement with our experimental result (0.5 g TKX-50 is dissolved in 45 mL formic acid in this work). Note that due to PMF corresponding to the dimensions (3 × 3 × 3), instead of one-dimensional reaction coordinates, the values of PMF are higher than the real free-energy barrier. Moreover, the differences in PMF between the transition state and TKX-50 are close to each other, and those between the transition state and the supersaturated formic acid solution are also close to each other. This indicates that the difference is not related to the type of OPs or whether it is K-means clustering, as is in agreement with our previous investigation [56].

#### 3.3. MD Simulation of Crystallization of TKX-50 under the External Electric Fields

^{8}V/m are shown in Table 2. Compared with the values without the external electric fields, both the peak values and the concentration parameters of the pair distribution functions for TKX-50 crystal change significantly, especially for them under the external electric field along the direction of the c-axis. As expected, due to two hydroxyamine groups having a positive charge and the dihydroxy-5,5’-tetrazolium group having a negative charge, they will undergo significant displacement under the external electric field, resulting in a significant change in the peak values and the concentration parameters, as shown in Figure 1c. Most of the directions of the positive and negative charges for hydroxylamine and dihydroxy-5,5’-tetrazolium groups are exactly along the direction of the c-axis of the crystal (see Figure 1h), so the changes are more significant along the c-axis. Moreover, in most cases, the values of ${\varphi}_{q}$ have decreased while those of ${\varphi}_{\widehat{r}}^{\alpha}$ increased, showing that the π∙∙∙π stacking is more significant under the external electric field.

^{8}V/m by K-means clustering. Compared with those without the external electric field (see (IV)), the values of the ln(Convergence) of the relative orientation OPs are decreased greatly, and there is convergence after 60 iterations in all cases. This shows that the external electric field can accelerate the evolution of the string. In 2017, Jha et al. found that external electric fields can obviously increase the nucleation rate [59]. Koizumi et al. also observed the same phenomenon [73]. These results confirmed that, shown by the accelerated evolution of the string, the external electric field can accelerate the nucleation rate.

^{−1}under the external electric field. The difference in PMF between the TKX-50 crystal and the transition state is less than 40.0 kJ∙mol

^{−1}under the external electric fields with a strength of 51.40 × 10

^{8}V/m along the a-, b-, and c-axes. These values are decreased in comparison with those in no field. Moreover, the fields parallel to the b- and c-axes affect the difference in PMF between the TKX-50 crystal and the transition state less than those parallel to the a-axis. The crystallization of TKX-50 from the formic acid solution is mainly closely related to the intermolecular interaction. From Figure 1f–h, along the a-axis, the strong H-bonding interactions are formed. They are strengthened by the increased dipole moments more significantly under the external electric fields, leading to a large value of the PMF between the TKX-50 crystal and the transition state along the a-axis. The value of the PMF between them is the smallest along the b-axis. From Figure 1f–h, the intermolecular π∙∙∙π stackings are formed along the b-axis. Due to the weaker π∙∙∙π stacking than H-bonding interaction, the more notable change will occur under the external electric field along the b-axis, leading to the low energy of the transition state and thus the smaller difference in PMF. Therefore, applying an external electric field along the b-axis direction has more practical value for achieving the crystallization of TKX-50 from solution.

^{8}, 5.14 × 10

^{8}, and 10.28 × 10

^{8}~102.80 × 10

^{8}V/m with a step of 10.28 × 10

^{8}V/m. The values of the difference in PMF between the TKX-50 crystal and the transition state were calculated to be 51.3, 48.2, 37.6, 42.5, 40.1, 38.3, 36.6, 35.3, 39.7, 42.5, 46.8, and 32.5 kJ/mol. The corresponding 3D surface plots of the free-energy landscape obtained from ${\theta}_{C}^{d}$ and ${\theta}_{C}^{b}$ as the collective variables are shown in Figure 8. The influence of the external electric field on PMF has no regular pattern in values. For the former, a good linear result (R

^{2}= 0.9882) was found, showing that the external electric fields (F) have an obvious influence on the polymorphic transformation from o-TNT to m-TNT. For the latter, this relationship was not discovered from the above values of the difference in PMF between the TKX-50 crystal and the transition state. As mentioned above, we only found that the values of the difference in PMF are decreased in comparison with those in no field. It is well-known that the movement of ions in an external electric field is complex, and in the process of the crystallization from solution, this movement pattern may be more complex, leading to irregular changes in the difference in PMF. However, one conclusion can be confirmed that the external electric fields can reduce the difference in PMF and thus accelerate the nucleation rate, as is also confirmed by the experimental result [59].

#### 3.4. Detonation Performance

^{3}, see Table S1 without field vs. 1.933 g/cm

^{3}, see Table S6 under external electric field), the calculated detonation velocity (V

_{D}) and detonation pressure (P

_{D}) were 9604 m/s and 42.49 GPa in no field, and 9650 m/s and 43.07 GPa with an external electric field, respectively. The results of V

_{D}and P

_{D}without field are consistent with those in Reference [74]. Compared with the values without the external electric fields, the values of V

_{D}and P

_{D}under the external electric field were increased by 0.48% and 1.37%, respectively. This indicates that the external electric field can enhance the detonation performance of TKX-50.

## 4. Experiment and MD Simulation Details

#### 4.1. Experiment

#### 4.2. MD Simulation Details

_{3}rotation, the internal degrees of freedom of conformational transformation can be ignored. Based on molecular symmetry and point molecule representation [69], The vector r joins the center of mass of the two tetrazolium rings. The vector normal to the plane of the tetrazolium ring gives the absolute orientation q of each of the tetrazolium anions. The bond orientation ${\varphi}_{\widehat{r}}$ is given by the angle formed by the vectors r and q1, and the relative orientation ${\varphi}_{q}$ is given by the angles between the vectors q1 and q2, as shown in Figure 2.

^{−1}was introduced for the temperature control. The Nosé–Hoover–Langevin piston with a damping time of 50 fs was used to control the pressure. The hydrogen bond lengths were constrained with the LINCS algorithm [76]. Periodic boundary conditions were applied in all directions. After equilibrium, annealing treatment was carried out at a constant volume to zero temperature, and energy minimization relaxation was carried out. Then, the temperature was raised to 300 K, and a 2.5 ns MD simulation was carried out at a constant volume with a time step of 0.5 fs. Based on the obtained trajectory, the parameters of Equations (1)–(7) were calculated using the maximum-likelihood estimation method, and the initial order parameters were determined.

^{8}V/m were added along the positive direction of the a-, b-, and c-axes. For the c-axis, the field strengths are 0.00, 0.514 × 10

^{8}, 5.14 × 10

^{8}, and 10.28 × 10

^{8}~102.80 × 10

^{8}V/m with a step of 10.28 × 10

^{8}V/m were also applied. The MD simulation of the structural optimization was carried out by Materials Studio 5.0 software package with COMPASS force field.

#### 4.3. Detonation Performance Calculations

_{D}) and detonation pressure (P

_{D}) can be evaluated by Kamlet approximation, as shown by Equation (22) [78]:

_{D}and d are the heat of detonation reaction and density of explosives, respectively.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Experimental setup of DC external electric field and crystallization of TKX-50. (

**a**): Experimental setup; (

**b**): Schematic diagram of anion motion in an external electric field; (

**c**):Molecular structure of TKX-50 obtained from experiment; (

**d**): Molecular stacking in crystals; (

**e**): Perspective of molecular stacking from the BC plane direction of the crystal (

**f**–

**h**): The directions of the external electric fields along the three axes. Note: In (

**f**–

**h**), a, b, and c mean the a-axis, b-axis, and c-axis, respectively.

**Figure 2.**XRD powder diffraction of TKX-50. (

**a**): XRD under external electric field; (

**b**): XRD without external electric field.

**Figure 3.**Construction of OPs for TKX-50. Red, blue, gray, and white represent O, N, C, and H atoms, respectively. The vector

**r**joins the center of mass of the two molecules. The direction of the axis across the center of the ring can be used as an approximate measure of the absolute orientation (q

_{1}or q

_{2}). The bond orientation ${\varphi}_{\widehat{r}}$ is defined as the projection of ${\widehat{r}}_{ij}$ onto q

_{1}, and the relative orientation ${\varphi}_{q}$ shows the rotation of q

_{1}onto q

_{2}.

**Figure 4.**The distribution of (

**a**) distance, (

**b**) bond orientation, and (

**c**) relative orientation OPs for TKX-50 crystal and the corresponding forms in the formic acid solution. These distributions were obtained by considering 2.5 ns MD simulations in the NPT ensemble from the averaged values within the divided cells with the 3 × 3 × 3 grid for each of the systems with 216 TKX-50 molecules.

**Figure 5.**Convergence of collective variables during the evolution of the string. (I) and (II) correspond to ${\varphi}_{C}^{db}$ with K-means and without K-means clustering, respectively; (III) and (IV) correspond to ${\varphi}_{C}^{dr}$ without K-means clustering and with K-means clustering, respectively. (V), (VI), and (VII) correspond to ${\varphi}_{C}^{dr}$ with K-means clustering for the convergence of collective variables under an external electric field with the strength of 51.40 × 10

^{8}V/m along the a-, b-, and c-axes, respectively. (

**a**) Convergence of collective variables without external electric field; (

**b**) convergence of collective variables under external electric field.

**Figure 6.**PMF as a function of arclength along the FTS path for the converged string obtained from the SMCV. The initial point at arclength zero is the supersaturated formic acid solution of TKX-50, and the endpoint is TKX-50 crystal. (

**I**,

**II**) correspond to the FTS path obtained from ${\varphi}_{C}^{db}$ with K-means and without K-means clustering; (

**III**,

**IV**) correspond to the FTS path obtained from ${\varphi}_{C}^{dr}$ without K-means clustering and with K-means clustering; (

**V**–

**VII**) correspond to the FTS path obtained from ${\varphi}_{C}^{dr}$ with K-means clustering for the convergence of collective variables under external electric field with the strength of 51.40 × 10

^{8}V/m along the a-, b-, and c-axes, respectively. In every image, B means the transition state, and A and C mean the state similar to the solution and the crystal, respectively.

**Figure 7.**Changes in local order parameters on the FTS path shown by the key snapshots. Along the (

**a**–

**l**), the type of molecule gradually changes from the “all blue” initial conformation, the formation of molecular clusters of which “Scattered green areas are surrounded by blue areas”, to gradually increasing “sporadic green” molecular clusters, and then to the “blue being engulfed by green” conformation. (

**a**–

**l**)

**mean the several**key snapshots from crystal to supersaturated formic acid solution of TKX-50 (the molecules of formic acid were deleted).

**Figure 8.**Three-dimensional surface plots of the free-energy landscape obtained from ${\theta}_{C}^{d}$ and ${\theta}_{C}^{b}$ as the collective variables for the crystallization of TKX-50 from the supersaturated formic acid solution under the external electric fields along the b-axis: (

**a**) 10.28 × 10

^{8}V/m; (

**b**) 20.56 × 10

^{8}V/m; (

**c**) 30.84 × 10

^{8}V/m; (

**d**) 41.12 × 10

^{8}V/m; (

**e**) 51.40 × 10

^{8}V/m; (

**f**) 61.68 × 10

^{8}V/m; (

**g**) 71.96 × 10

^{8}V/m; (

**h**) 82.24 × 10

^{8}V/m.

r (Å) | 1/σ^{2} (Å^{−1}) | ${\mathit{\varphi}}_{\widehat{\mathit{r}}}^{\mathit{\alpha}}$ (°) | ${\mathit{\eta}}_{\widehat{\mathit{r}}}^{\mathit{\alpha}}$ | ${\mathit{\varphi}}_{\mathit{q}}$ (°) | ${\mathit{\eta}}_{\mathit{q}}^{\mathit{\alpha}}$ |
---|---|---|---|---|---|

5.53 (5.46) | 28.56 | 11.53 (12.68) | 31.25 | 75.31 (69.68) | 11.26 |

7.56 (7.62) | 5.19 | 69.82 (67.53) | 28.62 | 43.26 (44.52) | 23.18 |

8.69 (8.43) | 16.40 | 78.93 (83.26) | 10.63 | 14.83 (16.83) | 17.53 |

^{a}The values in parentheses are the experimental results.

**Table 2.**Average peak locations and concentration parameters for the TKX-50 crystal at 300 K under the external electric field with the strength of 51.40 × 10

^{8}V/m.

r (Å) | 1/σ^{2} (Å^{−1}) | ${\mathit{\varphi}}_{\widehat{\mathit{r}}}^{\mathit{\alpha}}$ (°) | ${\mathit{\eta}}_{\widehat{\mathit{r}}}^{\mathit{\alpha}}$ | ${\mathit{\varphi}}_{\mathit{q}}$ (°) | ${\mathit{\eta}}_{\mathit{q}}^{\mathit{\alpha}}$ |
---|---|---|---|---|---|

5.62 ^{a} (5.50) ^{b}6.03 ^{c} | 18.69 (26.50) 15.83 | 42.82 (61.29) 68.66 | 32.83 (18.39) 30.26 | 25.69 (27.57) 19.68 | 11.53 (12.63) 9.10 |

7.56 (7.16) 6.98 | 29.02 (10.31) 6.28 | 69.82 (66.53) 70.15 | 25.72 (26.17) 36.01 | 32.55 (11.53) 28.62 | 20.12 (16.93) 22.26 |

8.69 (8.53) 8.55 | 15.62 (23.19) 21.51 | 78.18 (82.31) 80.66 | 15.70 (18.63) 19.25 | 16.76 (15.92) 17.39 | 23.17 (9.13) 16.52 |

^{a}The values of the average peak locations and concentration parameters corresponding to the positive direction of the a-axis.

^{b}The values of the average peak locations and concentration parameters (in parentheses) corresponding to the positive direction of the b-axis.

^{c}The values of the average peak locations and concentration parameters (in italics) corresponding to the positive direction of the c-axis.

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## Share and Cite

**MDPI and ACS Style**

Ren, F.; Wang, X.; Zhang, Q.; Wang, X.; Chang, L.; Zhang, Z.
Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals. *Molecules* **2024**, *29*, 1159.
https://doi.org/10.3390/molecules29051159

**AMA Style**

Ren F, Wang X, Zhang Q, Wang X, Chang L, Zhang Z.
Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals. *Molecules*. 2024; 29(5):1159.
https://doi.org/10.3390/molecules29051159

**Chicago/Turabian Style**

Ren, Fude, Xiaolei Wang, Qing Zhang, Xiaojun Wang, Lingling Chang, and Zhiteng Zhang.
2024. "Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals" *Molecules* 29, no. 5: 1159.
https://doi.org/10.3390/molecules29051159